A least-squares/fictitious domain method for incompressible viscous flow around obstacles with Navier slip boundary condition

Abstract

In this article, we discuss a least-squares/fictitious domain method for incompressible viscous flow around obstacles with Navier slip boundary condition. Assuming that Ω and B are two bounded sub-domains of Rd, with B‾⊂Ω, in order to solve the incompressible Navier–Stokes equations with a Navier slip condition on the boundary γ of the obstacle B, we advocate a fictitious domain method where one solves a simpler variant of the original problem on the whole Ω, followed by a well-chosen correction over B. This method is of the virtual control type and relies on a least-squares formulation making the problem solvable by a conjugate gradient algorithm operating in a well chosen control space. A detailed discussion of the finite element implementation of the above methodology is also provided. Numerical results are given; they suggest optimal order of convergence.